login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A116549 a(0) = 1. a(m + 2^n) = a(n) + a(m), for 0 <= m <= 2^n - 1. 4
1, 2, 3, 4, 4, 5, 6, 7, 5, 6, 7, 8, 8, 9, 10, 11, 5, 6, 7, 8, 8, 9, 10, 11, 9, 10, 11, 12, 12, 13, 14, 15, 6, 7, 8, 9, 9, 10, 11, 12, 10, 11, 12, 13, 13, 14, 15, 16, 10, 11, 12, 13, 13, 14, 15, 16, 14, 15, 16, 17, 17, 18, 19, 20 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Consider the following bijection between the natural numbers and hereditarily finite sets. For each n, write out n in binary. Assign to each set already given a natural number m the (m+1)-th digit of the binary number (reading from right to left). Let the set assigned to n contain all and only those sets which have a 1 for their digit. Then a(n) gives the number of pairs of braces appearing in the n-th set written out in full, e.g., for 3, we have {{{}}{}}, with 4 pairs of braces. - Thomas Anton, Mar 16 2019

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

FORMULA

For n > 0: a(n) = a(A000523(n)) + a(A053645(n)). - Reinhard Zumkeller, Aug 27 2014

EXAMPLE

From Gus Wiseman, Jul 22 2019: (Start)

A finitary (or hereditarily finite) set is equivalent to a rooted identity tree. The following list shows the first few rooted identity trees together with their corresponding index in the sequence (o = leaf).

   0: o

   1: (o)

   2: ((o))

   3: (o(o))

   4: (((o)))

   5: (o((o)))

   6: ((o)((o)))

   7: (o(o)((o)))

   8: ((o(o)))

   9: (o(o(o)))

  10: ((o)(o(o)))

  11: (o(o)(o(o)))

  12: (((o))(o(o)))

  13: (o((o))(o(o)))

  14: ((o)((o))(o(o)))

  15: (o(o)((o))(o(o)))

  16: ((((o))))

  17: (o(((o))))

  18: ((o)(((o))))

  10: (o(o)(((o))))

(End)

MATHEMATICA

Nest[Append[#1, #1[[#3 + 1]] + #1[[#2 - 2^#3 + 1]] & @@ {#1, #2, Floor@ Log2@ #2}] & @@ {#, Length@ #} &, {1}, 63] (* Michael De Vlieger, Apr 21 2019 *)

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];

dab[n_]:=1+Total[dab/@(bpe[n]-1)];

Array[dab, 30, 0] (* Gus Wiseman, Jul 22 2019 *)

PROG

(Haskell)

import Data.Function (on); import Data.List (genericIndex)

a116549 = genericIndex a116549_list

a116549_list = 1 : zipWith ((+) `on` a116549) a000523_list a053645_list

-- Reinhard Zumkeller, Aug 27 2014

CROSSREFS

Cf. A000523, A053645.

Cf. A000081, A000120, A004111, A029931, A048793, A061775, A070939, A072639, A276625, A279861, A326031.

Sequence in context: A234604 A236346 A306890 * A268382 A107079 A025528

Adjacent sequences:  A116546 A116547 A116548 * A116550 A116551 A116552

KEYWORD

easy,nonn

AUTHOR

Leroy Quet, Mar 16 2006

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 18 04:50 EDT 2019. Contains 326072 sequences. (Running on oeis4.)